Suspension system with optimized damper response for wide range of events

ABSTRACT

An analytical methodology for the specification of progressive optimal compression damping of a damper of a suspension system to negotiate a multiplicity of severe events, yet provides very acceptable ride quality and handling during routine events. The damping response of the damper is optimized based upon a progressive optimal constrained events damping function derived from a low envelope curve incorporated with a predetermined damper force acting on the wheel center below a predetermined wheel center velocity, u 1 , based on ride and handling considerations for a given vehicle or vehicle model according to the prior art methodology, whereby the low envelope curve is constructed utilizing a one degree of freedom nonlinear mechanical system model or a quarter car nonlinear mechanical system model.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application invention is a continuation-in-part of patentapplication Ser. No. 11/939,698, filed on Nov. 14, 2007, whichapplication is presently pending, and which application and the presentapplication claim the priority benefit of provisional patent applicationSer. No. 60/967,209, filed on Aug. 31, 2007 which has now passed its oneyear pendancy term.

TECHNICAL FIELD

The present invention relates to motor vehicle suspension systems,wherein the motor vehicle body is sprung in relation to each of itswheels via a respective spring-damper combination. More particularly,the present invention relates to a method for providing a suspensionsystem with optimized damper response based upon the damper beingadjusted in accordance with a progressive optimal constrained eventsdamping function applicable to a multiplicity of jounce events,including those involving maximum wheel displacements

BACKGROUND OF THE INVENTION

Motor vehicle suspension systems are configured so that the wheels areable to follow elevational changes in the road surface as the vehicletravels therealong. When a rise in the road surface is encountered, thesuspension responds in “jounce” in which the wheel is able to moveupwardly relative to the frame of the vehicle. On the other hand, when adip in the road surface is encountered, the suspension responds in“rebound” in which the wheel is able to move downwardly relative to theframe of the vehicle. In either jounce or rebound, a spring (i.e., coil,leaf, torsion, etc.) is incorporated at the wheel in order to provide aresilient response to the respective vertical movements with regard tothe vehicle frame. However, in order to prevent wheel bouncing andexcessive vehicle body motion, a damper (i.e., shock absorber, strut,etc.) is placed at the wheel to dampen wheel bounce. Additionally, whenthe limit of jounce is encountered, it is customary to provide a maximumjounce impact absorber in the form of a bumper cushion.

Referring now to FIGS. 1 through 1B, components of a conventionalsuspension system 10 are depicted which allow for jounce and rebound ata wheel of the subject motor vehicle 12.

Firstly with regard to FIG. 1, a control arm 14 is pivotally mountedwith respect to the frame 16, wherein, in the depicted example, atorsion spring 18 is utilized to provide resilient response for thejounce and rebound of the control arm relative to the frame. To providecontrol over the rate of jounce and rebound, a damper in the form of ashock absorber 20 is connected pivotally at one end to the frame 16 andconnected pivotally at the other end to the control arm 14.Alternatively, a damper in the form of a strut may be used in thesuspension system, as for example disclosed in U.S. Pat. No. 5,467,971.To provide cushioning in the event a maximum jounce occurs, a jouncebumper cushion 22 is mounted to the frame 16 which is resilientlycompressed by movement of the control arm as jounce approaches itsmaximum.

Referring next to FIG. 1A, the internal components and operationalaspects of a conventional shock absorber 20′ (a remote reservoir highpressure gas type shock absorber being shown merely by way of example)can be understood. A valved piston 30 is reciprocably movable within ashock cylinder 32. A shock rod 34 is attached to the valved piston 30and is guided by a shock rod guide 36 at one end of the shock cylinder32. Below the valved piston 30 and above the shock rod guide 36 is amutually interacting rebound limiter 38. The instantaneous position ofthe valved piston 30 within the shock cylinder 32 defines a firstinterior portion 32F and a second interior portion 32S of the interiorof the shock cylinder. In the example depicted at FIG. 1A, thepressurization in the first and second interior portions 32F, 32S isprovided by an hydraulic fluid O which is pressurized by pressurizedgas, preferably nitrogen, G acting on a divider piston 40 of anhydraulic fluid reservoir cylinder 42, wherein a tube 44, including abase valve 44V, connects the hydraulic fluid between the hydraulic fluidreservoir cylinder and the first interior portion. In operation, as thecontrol arm undergoes jounce, the hydraulic fluid is displaced from thefirst interior portion into the hydraulic fluid reservoir cylinder,causing the pressure of the nitrogen gas to increase as its volumedecreases and thereby causing an increased hydraulic pressure on thevalved piston 30 in a direction toward the shock rod guide. Hydraulicfluid is able to directionally meter through valving 46 of the valvedpiston 30 in a manner which provides damping.

Referring next to FIG. 1B, the internal structure of a conventionaljounce bumper cushion 22 can be understood. An optional skin 50 of acompliant material (i.e., having energy absorbing or damping properties)may, or may not, overlay an interior of resilient elastomeric material52, which may be for example a rubber, rubber-like material, ormicro-cellular urethane. In operation as the control arm approachesmaximum jounce, the jounce bumper cushion 22 compresses, delivering areaction force on the control arm which increases with increasingcompression so as to minimize the severity of impact of the control armwith respect to the frame at the limit of jounce. Immediately followingthe jounce, the rebound involves the energy absorbed by the compressionof the conventional bumper cushion being delivered resiliently back tothe suspension.

In the art of motor vehicle suspension systems, it is known that aconventional jounce bumper cushion and related dampers can show wear. Itis also known that when the energy absorbed from a particular bump ordip exceeds the capacity of a conventional jounce bumper cushion, a hardmechanical stop is engaged. This abrupt transfer of jounce force andenergy to the frame manifests itself in the passenger compartment as asharp jolt, which can create load management issues in addition to thediscomfort of a rough ride. Further, in order for the frame to acceptsuch impact loads, the structure of the frame must be engineered for anappropriate strength, which is undesirable from the standpoint of theadded vehicle weight such structures must inherently entail.

Vehicle suspension engineering has traditionally focused on ride andhandling as this pertains to body and wheel relative motion with respectto the body below about 1.3 m/s (meters per second). However, thesuspension travel requirements in a vehicle are mainly driven by severeevents which generate maximum displacements of the wheel relative to thebody. These severe events, such as when the vehicle encounters a deepand steep-walled pothole, can generate wheel velocities (relative to thebody) of up to 9 m/s.

An approach pursued by Bavarian Motor Works (BMW) of Munich, Germany, isdescribed in European Patent Application EP 1,569,810 B1, published onSep. 7, 2005; which application is parent to U.S. Patent Applicationpublication 2006/0243548 A1, published on Nov. 2, 2006.

The object of the BMW disclosure of EP 1,569,810 B1 is to provide avibration damping method on a motor vehicle wheel suspension by means ofa hydraulic vibration damper which prevents great loads on the vehiclebody and chassis caused by very large vertical velocities of the wheel,e.g., when traveling over potholes. According to the BMW disclosure, ina hydraulic vibration damper for a motor vehicle, a method of vibrationdamping on a wheel suspension is used by BMW, characterized in that thedamping force of the vibration damper increases as a function of pistonspeed, especially in the piston speed range of essentially 0 to 2 m/s,at first increasing slowly, essentially linearly, and then, especiallyabove a piston speed of essentially 2 m/s, increasing according to ahighly progressive function. Further according to the BMW disclosure,through a suitable choice, design and construction of vibration dampervalves or by otherwise influencing the hydraulic resistances in thevibration damper, it is possible to implement a characteristic which isgenerated by damping forces known from the state of the art in thepiston speed range up to the end of the range that is relevant forcomfort, and beyond this piston speed range, an extreme progression inthe damper characteristic is induced to decelerate the acceleratedmasses to a greater extent.

While the BMW disclosure seeks to provide a solution to thelong-standing problem of damping excessively large wheel-to-bodyvelocities while attempting to maintain acceptable ride and handling forlow velocities, the disclosure requires an ad hoc reliance upon apresupposed and essential damper curve which is devoid of any underlyingphysics which supports any of the curve aspects. Thus, what yet remainsneeded in the art is an analytical methodology to predict damping curveswhich truly achieve the goal of damping excessively large wheel-to-bodyvelocities while attempting to maintain acceptable ride and handling forlow velocities.

Of additional note is Japan Society of Automotive Engineers, JSAEtechnical paper 9306714 by Miyazaki, Kiyoaki, Yasai, Hirofumi, “A studyof ride improvement of the bus”, JSAE Autumn Convention Nagoya, JapanOct. 19-21, 1993, wherein the authors confirmed that a progressivedamping characteristic is effective for reducing the pitching and impactvibration.

Of further note is Society of Automotive Engineers, SAE technical paper2006-01-1984 by Benoit Lacroix, Patrice Seers and Zhaoheng Liu, “APassive Nonlinear Damping Design for a Road Race Car Application”,wherein a nonlinear passive damping design is proposed to optimize thehandling performance of an SAE Formula car in terms of roll and pitchresponses.

Progressive damping is thought of as an enabler to reduce harsh impact,ride input feel when encountering severe events through the method ofmaintaining a predefined load in jounce and reducing engagement into thejounce suspension stop. It is also needed to develop enablers to reducetotal jounce travel so that a given vehicle could be trimmed lower toenable competitive styling cues. Trimming a vehicle lower usuallyincreases the level of harshness for an event such as a deep pothole andother severe events.

What remains needed in the art, therefore, is an analytical methodologyfor the specification of a progressive optimal constrained eventsdamping function enabling a motor vehicle suspension system to negotiatea multiplicity of severe events, such as a multiplicity of potholes,with reduced harshness, yet provides very acceptable ride quality andhandling during routine events, such as common road surfaces, limitspeak loads on the fame structure, reduces wheel travel, and enableslower trim height.

SUMMARY OF THE INVENTION

The present invention is an analytical methodology for the specificationof a progressive optimal constrained events damping function enabling amotor vehicle suspension system to negotiate a multiplicity of severeevents, such as a multiplicity of potholes, with reduced harshness, yetprovides very acceptable ride quality and handling during routineevents, such as common road surfaces, limits peak loads on the framestructure, reduces wheel travel, and enables lower trim height.

A method to provide a progressive optimal unconstrained event dampingfunction of the wheel assembly with respect to the body, employing a onedegree of freedom (1DOF) nonlinear mechanical system model, is generatedfrom equations of motion of the wheel center of the wheel assembly withno initial external forces, no initial displacement, and the total forceacting on the wheel center is essentially constant (hereafter referredto simply as “constant total force”) during the wheel center'sdeceleration from an initial velocity U₀ to a velocity of zero. Theconstant total force is related to a determined travel length of thewheel center such that when the wheel center is at the determined travellength, its velocity is zero, the damper force is zero and thesuspension spring is compressed the determined travel length by whichthe suspension spring force is equal to the constant total force andwhen the wheel center is at zero displacement its velocity is U₀, thesuspension spring is uncompressed with respect to equilibrium by whichthe suspension spring force is zero, and the damper force is equal tothe constant total force. With the above conditions, the amount ofenergy dissipated by the damper is maximized and the total load on thebody is minimized, whereby a progressive optimal unconstrained eventdamping function is obtained which is valid for all displacements of thewheel center from zero to the predetermined travel length and velocitiesfrom U₀ to zero.

The suspension spring may include coil spring, jounce bumper, mounts,and other suspension compliances. Suspension spring force as a functionof wheel center travel can be determined in the lab through the standardtechnique, where the tire patches are actuated vertically in jounce andrebound while the force is measured through the force tables and wheeltransducer systems.

In practice, a predetermined damper force acting on the wheel centerbelow a wheel center velocity u₁, approximately 2.0 m/s, is based onride and handling considerations for a given vehicle or vehicle modelaccording to the prior art methodology, and should not be alteredtherefrom.

A method to provide a progressive optimal constrained event dampingfunction of the wheel assembly with respect to the body, employing a onedegree of freedom (1DOF) nonlinear mechanical system model, depicted inFIG. 2, in which the predetermined damper force acting on the wheelcenter below or equal to a wheel center velocity of u₁, approximately2.0 m/s, is not altered, is generated as described below:

1. A progressive optimal constrained damper force is obtained fromequations of motion of the wheel center with no initial external forces,an initial displacement x₀ when the initial velocity is U₀, and thetotal force acting on the wheel center is constant during the wheelcenter's deceleration from a velocity U₀ to an empirically determinedvelocity u₂. The constant total force acting on the wheel center isrelated to equations of motion of the wheel center and predeterminedvehicle parameters.

2. A smooth, continuous damping force transition function is obtained,preferably approximating a step function, producing a damping force fromthe wheel center velocity u₁ to an empirically determined wheel centervelocity u₂ greater than u₁, but neighboring, u₁.

3. The predetermined damper force acting on the wheel assembly is usedbelow or equal to a wheel assembly velocity u₁.

The constant total force is related to a determined travel length of thewheel center such that when the wheel center is at the determined travellength, its velocity is zero, the damper force is zero and thesuspension spring is compressed the determined travel length by whichthe suspension spring force is equal to the constant total force andwhen the wheel center is at displacement x₀ its velocity is U₀, thesuspension spring is compressed by x₀.

With the above conditions, the amount of energy dissipated by the damperis maximized and the total load on the body is minimized whereby aprogressive optimal constrained event damping function is obtained validfor all displacements of the wheel center from zero to the determinedtravel length and velocities from U₀ to zero.

However, a first progressive optimal constrained event damping functionvalid for all displacements of the wheel center from zero to thedetermined travel length and velocities from a first initial velocityU₀₁, associated with a first event such as a first pothole, to zero willnot be an optimal constrained event damping function valid for alldisplacements of the wheel center from zero to the determined travellength and velocities from a second initial velocity U₀₂, associatedwith a second event such as a second pothole, to zero. Each wheel centervelocity event has a progressive optimal constrained event dampingfunction depending on the peak initial wheel center velocity, but theoptimal constrained event damping function for one wheel center velocityevent is not an optimal constrained event damping function for anotherwheel center velocity event having a different peak initial wheel centervelocity. A progressive optimal constrained event damping functionobtained for a given predetermined initial wheel center velocity U₀ forall displacements of the wheel center from zero to the determined travellength and velocities from U₀ to zero is not an optimal constrainedevents damping function for a multiplicity of peak initial wheel centervelocities.

A preferred aspect of the present invention is a progressive optimalconstrained events damping function enabling a motor vehicle suspensionsystem to negotiate a multiplicity of severe events, such as amultiplicity of potholes, with reduced harshness, yet provides veryacceptable ride quality and handling during routine events, such ascommon road surfaces, limits peak loads on the frame structure, reduceswheel travel, and enables lower trim height. This progressive optimalconstrained events damping function is a low envelope curve incorporatedwith the predetermined damper force acting on the wheel center below awheel center velocity u₁, approximately 2.0 m/s, based on ride andhandling considerations for a given vehicle or vehicle model accordingto the prior art methodology, as previously described, whereby the 1DOFnonlinear mechanical system model progressive optimal constrained eventdamping functions are utilized to construct the low envelope curve.

The low envelope curve according to the preferred aspect of the presentinvention is generated from a plot consisting of a multiplicity ofprogressive optimal constrained event damping functions utilizing the1DOF nonlinear mechanical system model by constructing a curve passingthrough the peak initial damper velocities, from the highest peakinitial damper velocity to the lowest peak initial damper velocity,successively, to a predetermined wheel center velocity u₁, approximately2.0 m/s, thereafter, whereat a predetermined damper force acting on thewheel center is based on ride and handling considerations for a givenvehicle or vehicle model according to the prior art methodology, aspreviously described. This low envelope curve is incorporated with thepredetermined damper force acting on the wheel center below a wheelcenter velocity u₁, approximately 2.0 m/s, based on ride and handlingconsiderations for a given vehicle or vehicle model according to theprior art methodology, as previously described, to generate theprogressive optimal constrained events damping function according to thepreferred aspect of the present invention.

This progressive optimal constrained events damping function accordingto the preferred aspect of the present invention may not be optimal fora particular given peak initial wheel center velocity U₀ but there is noother events damping function that will not increase the load on thesprung mass for a multiplicity of peak initial wheel center velocities.That is, this progressive optimal constrained events damping functionwill not increase the load for any given peak initial wheel centervelocity U₀ of a multiplicity of peak initial wheel center velocities.

In practice, a quarter car nonlinear mechanical system model, depictedin FIG. 10, is used in the art for suspension design. A most preferredaspect of the present invention employs the quarter car nonlinearmechanical system model of FIG. 10 to obtain a progressive optimalconstrained events damping function valid for all displacements of thewheel center from zero to the determined travel length for multiple peakinitial wheel center velocities to zero utilizing quarter care modelprogressive optimal constrained event damping functions. However, ananalytical solution utilizing the quarter car nonlinear mechanicalsystem model to obtain a progressive optimal constrained event dampingfunction valid for all displacements of the wheel center from zero tothe determined travel length for a given peak initial wheel centervelocity from U₀ to zero is not available and numerical techniques areused. The 1DOF nonlinear mechanical system model analytical progressiveoptimal constrained event damping function for a given peak initialwheel center velocity, as previously described, is utilized as theinitial function for the quarter car nonlinear mechanical system modelto obtain a progressive optimal constrained event damping function validfor all displacements of the wheel center from zero to the determinedtravel length for a given peak initial wheel center velocity from U₀ tozero.

The most preferred aspect of the present invention is a progressiveoptimal constrained events damping function enabling a motor vehiclesuspension system to negotiate a multiplicity of severe events, such asa multiplicity of potholes, with reduced harshness, yet provides veryacceptable ride quality and handling during routine events, such ascommon road surfaces, limits peak loads on the frame structure, reduceswheel travel, and enables lower trim height. This progressive optimalconstrained events damping function is a low envelope curve incorporatedwith the predetermined damper force acting on the wheel center below awheel center velocity u₁, approximately 2.0 m/s, based on ride andhandling considerations for a given vehicle or vehicle model accordingto the prior art methodology, as previously described, whereby thequarter car nonlinear mechanical system model progressive optimalconstrained event damping functions are utilized to construct the lowenvelope curve.

The low envelope curve according to the most preferred aspect of thepresent invention is generated from a plot consisting of a multiplicityof progressive optimal constrained event damping functions utilizing thequarter car nonlinear mechanical system model by constructing a curvepassing through the peak initial damper velocities, from the highestpeak initial damper velocity to the lowest peak initial damper velocity,successively, to a predetermined wheel center velocity u₁, approximately2.0 m/s, thereafter, whereat a predetermined damper force acting on thewheel center is based on ride and handling considerations for a givenvehicle or vehicle model according to the prior art methodology, aspreviously described. This low envelope curve is incorporated with thepredetermined damper force acting on the wheel center below a wheelcenter velocity u₁, approximately 2.0 m/s, based on ride and handlingconsiderations for a given vehicle or vehicle model according to theprior art methodology, as previously described, to generate theprogressive optimal constrained events damping function according to themost preferred aspect of the present invention.

This progressive optimal constrained events damping function accordingto the most preferred aspect of the present invention may not be optimalfor a particular given peak initial wheel center velocity U₀ but thereis no other events damping function that will not increase the load onthe sprung mass for a multiplicity of peak initial wheel centervelocities. That is, this progressive optimal constrained events dampingfunction will not increase the load for any given peak initial wheelcenter velocity U₀ of a multiplicity of peak initial wheel centervelocities.

Accordingly, it is an object of the present invention to provide ananalytical methodology for the specification of a progressive optimalconstrained events damping function that enables a motor vehiclesuspension system to negotiate a multiplicity of severe events, such asa multiplicity of potholes, with reduced harshness, yet provides veryacceptable ride quality and handling during routine events, such ascommon road surfaces, limits peak loads on the frame structure, reduceswheel travel, and enables lower trim height.

This and additional objects, features and advantages of the presentinvention will become clearer from the following specification of apreferred embodiment.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view of a conventional motor vehicle suspensionsystem, including a control arm, a frame, a spring, a conventional shockabsorber and a conventional bumper cushion.

FIG. 1A is a sectional view of a conventional shock absorber.

FIG. 1B is a sectional view of a conventional bumper cushion.

FIG. 2 is a diagrammatic view of a 1DOF motor vehicle suspension systemdepicting a sprung mass (i.e., vehicle body), unsprung mass (i.e., wheelassembly), a spring, a damper, and a jounce bumper of a motor vehicle.

FIG. 3 is a graph of a plot of suspension spring normal force versuswheel center displacement for a representative motor vehicle.

FIG. 4 is a graph of progressive optimal unconstrained damping force atthe wheel center versus wheel center vertical velocity for therepresentative motor vehicle of FIG. 3.

FIG. 5 is a graph of damper force versus damper velocity for therepresentative motor vehicle of FIG. 3, showing a first plot of dampingfor a conventional passive damper, and a second plot of progressiveoptimal unconstrained damping.

FIG. 6 is a flow chart of an algorithm for a progressive optimalunconstrained damper force.

FIG. 7A is a flow chart of an algorithm to determine a constant totalforce for a progressive optimal constrained event damping function.

FIG. 7B is an example of a graph showing exemplar plots for carrying outthe algorithm of FIG. 7A.

FIG. 7C is a flow chart of an algorithm for the progressive optimalconstrained event damping function.

FIG. 8 is a graph of damper force versus damper velocity, showing afirst plot of damping for a conventional passive damper, and a secondplot of progressive optimal constrained damping, given the plot of FIG.3.

FIG. 9 is a graph of total suspension load versus time, showing a firstplot of a simulated suspension load having conventional passive damping;a second plot of a simulated suspension load having progressive optimalconstrained damping; and a third plot of a simulated suspension loadhaving progressive optimal unconstrained damping.

FIG. 10 is a diagrammatic view of a quarter car model of a motor vehiclesuspension system depicting a sprung mass (i.e., vehicle body), unsprungmass (i.e., wheel assembly), a first spring, a damper, a jounce bumperof a motor vehicle, a second spring (i.e., tire), and a road profile.

FIG. 11 is an exemplary diagram of potholes.

FIG. 12 is a graph of a plot of tire spring normal force versus wheelcenter displacement for a representative motor vehicle.

FIG. 13A is a flow chart of an algorithm to obtain a progressive optimalconstrained events damping function according to the preferred aspect ofthe present invention, utilizing a 1DOF nonlinear mechanical systemmodel.

FIG. 13B is a flow chart of an algorithm to obtain a progressive optimalconstrained events damping function according to the most preferredaspect of the present invention, utilizing a quarter car nonlinearmechanical system model.

FIG. 14 is an example of a graph of damper force versus damper velocity,depicting the construction of the low envelope curve according to thepreferred aspect of the present invention, utilizing a 1DOF nonlinearmechanical system model.

FIG. 15 is an example of a graph of damper force versus damper velocity,depicting the construction of the low envelope curve according to themost preferred aspect of the present invention, utilizing a quarter carnonlinear mechanical system model.

FIG. 16A is a graph of force versus time, showing a first plot ofsuspension load, a second plot of damper force, and a third plot oftotal force of a conventional passive damping system according to theprior art.

FIG. 16B is a graph of force versus time, showing a first plot ofsuspension load, a second plot of damper force, and a third plot oftotal force of a progressive optimal constrained events damping functionaccording to the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring now to the Drawing, FIGS. 2 through 16B depict various aspectsof the methodology to provide optimized damping in a motor vehiclesuspension system.

Generally speaking, the performance of motor vehicles under severe roadevents is tested using a pavement, which includes a series of potholes.For example, a minor pothole would be a shallow pit, and more pronouncedpothole would be a deeper pit capable of causing passengers to feel abounce; and a “severe event” pothole would be a box-shaped drop-off pitwith a hard, square edge at the back.

The following analysis is focused on motor vehicle suspension responseto traversal of a “severe event” pothole. During a “severe event”pothole traversal, the wheel first falls into the pothole, followed bythe falling body corner, and then, in an already jounced position(compared to nominal trim position), hits a steep bump approximating astep. Tire forces then accelerate the wheel and the suspension goes intoa deep jounce. Wheel vertical velocity reaches its peak, about 5 m/s,(MKS units being used herein) sometime through the jounce travel andthen decreases to zero at the maximum jounce travel (where the maximumshock tower vertical load is achieved). The deceleration portion of thejounce event (from the maximum wheel speed to zero) employs a one-degreeof freedom (1DOF) nonlinear mechanical system model, as described below.

FIG. 2 is a diagrammatic view 200 of a 1DOF motor vehicle suspensionsystem, typically used in the art, depicting the relationship ofpredetermined sprung mass 202 (i.e., the vehicle body), predeterminedunsprung mass m (i.e., the wheel assembly), nonlinear predeterminedspring 204, nonlinear damper 206 (i.e., shock absorber, etc.), andjounce bumper 208. Herein, the predetermined sprung mass 202 is referredto simply as the “body” wherein the body serves as the reference formeasuring velocity of the unsprung mass and the predetermined unsprungmass m is referred to simply as the “wheel assembly” 216.

In FIG. 2, the velocity, {dot over (x)}, or y (i.e., y={dot over (x)}),in the vertical direction x of the wheel center C_(w) of the wheelassembly 216 with respect to the body 202 is related to the velocity v,in the vertical direction x with respect to the body, of the bottom 214of damper 206 where it connects to the wheel assembly at point 212, by apredetermined ratio r such that y=v/r. Herein the velocity v is referredto as the damper velocity and the wheel center C_(w) is the centerlineof the wheel assembly 216.

The wheel assembly 216 is attached to the body 202 by the nonlinearpredetermined spring 204 and by the nonlinear damper 206 (the jouncebumper 208 is usually independently interfaced between the wheelassembly and the body). The displacement of the wheel center withrespect to the equilibrium position (nominal trim) 210 is in thevertical direction x and L is the travel length of the wheel center withrespect to the equilibrium position in the vertical direction x, whichcould include a portion of the jounce bumper 208, and also correspondsto the compression length of the predetermined spring 204. The travellength L is less than or equal to a predetermined maximum travel lengthL_(MAX) in the vertical direction x, as depicted merely by way ofexample in FIG. 2. The wheel assembly 216 and its mass m, predeterminedtravel length L, the predetermined maximum travel length L_(MAX), thespring 204, the body 202, and predetermined ratio r are empirically oranalytically determined for a particular vehicle or vehicle model by thevehicle manufacturer.

The equation of motion with no external forces acting on the wheelcenter C_(w) has the following form with the given initial conditions:m{umlaut over (x)}+F(x)+Φ({dot over (x)})=0, x(0)=x ₀ , {dot over(x)}(0)=U ₀   (1)wherein x is the displacement of the wheel center with respect to theequilibrium position 210, {dot over (x)} or y (i.e., y={dot over (x)})is the wheel center velocity with respect to the body 202, {umlaut over(x)} is the wheel center acceleration with respect to the body, Φ({dotover (x)}) is the damper force of the damper 206 as a function of wheelcenter velocity {dot over (x)}, F(x) is the suspension spring force ofthe spring 204 acting on the wheel center C_(w) at the displacement xcorresponding to a compression of the spring by a displacement x, x(0)is the position of the wheel center at time t=0 with respect to theequilibrium position 210, x₀ is the initial position of the wheel centerat time t=0 with respect to the equilibrium position 210, {dot over(x)}(0) is the velocity of the wheel center with respect to the body 202at time t=0, the travel length L is predetermined, and U₀ is apredetermined peak initial velocity of the wheel center with respect tothe body at time t=0 produced by a given predetermined “severe event”.In reality, suspension ride spring and damper are not collocated andwheel center vertical travel is not equal to the damper (shock)displacement. Given the predetermined ratio of damper (shock) travel perunit of vertical wheel center travel, r, y=v/r, wherein v is the damper(shock) velocity, and a predetermined initial damper velocity V₀, U₀ canbe calculated from U₀=V₀/r.

For the system 200 described by equation (1), assuming the velocity {dotover (x)}=y=0 when x=L≦L_(MAX), the suspension spring force F(x) of thespring 204 acting on the wheel center C_(w) is equal to F(L). If thetotal force, F(x)+Φ({dot over (x)}) acting on the wheel center C_(w)during its deceleration from U₀ to 0 is constant and equal to F(L), thenthe amount of energy dissipated by the damper 206 is maximized, and thetotal load on the body 202 is minimized. This leads to the followingcondition:F(x)+Φ(y)=F(L)=constant   (2)valid for 0≦x≦L≦L_(MAX), and 0≦y≦U₀ where Φ(y)=Φ({dot over (x)})represents a smooth, continuous, and monotonically increasingprogressive optimal unconstrained damper force of damper 206 as afunction of wheel center velocity y.

For initial conditions of the wheel center C_(w) being x(0)=x₀=0 and{dot over (x)}(0)=U₀, when the total force acting on the wheel centerduring its deceleration from a velocity of U₀ to 0 is constant and equalto F(L) and the progressive optimal unconstrained damper force Φ(y=0)=0,then the progressive optimal unconstrained damper force Φ(y) as afunction of wheel center velocity y of equation (2) can be expressed as:

$\begin{matrix}{{\Phi(y)} = {{F(L)} - {F\left( {\left( {1 - \frac{y^{2}}{U_{0}^{2}}} \right)L} \right)}}} & (3)\end{matrix}$whereby 0≦y≦U₀ and,

$\begin{matrix}{\frac{m \star U_{0}^{2}}{2} = {L \star {F(L)}}} & (4)\end{matrix}$which represents a kinetic energy constraint and wherein “*” representsa multiplication symbol.

The function

$F\left( {\left( {1 - \frac{y^{2}}{U_{0}^{2}}} \right)L} \right)$is the suspension spring force of the spring 204 acting on the wheelcenter C_(w) when the wheel center velocity is y where 0≦y≦U₀.

Since y=v/r and U₀=V₀/r, using equation (3), a progressive optimalunconstrained damper force Ψ₁(v) as a function of damper velocity v canbe expressed as:

$\begin{matrix}{{\Psi_{1}(v)} = \frac{\Phi\left( {y = {v/r}} \right)}{r}} & (5)\end{matrix}$or equivalently as:

$\begin{matrix}{{\Psi_{1}(v)} = {\frac{{F(L)} - {F\left( {\left( {1 - \frac{v^{2}}{V_{0}^{2}}} \right)L} \right)}}{r}.}} & (6)\end{matrix}$

The function

$F\left( {\left( {1 - \frac{v^{2}}{V_{0}^{2}}} \right)L} \right)$is the suspension spring force of the spring 204 acting on the wheelcenter C_(w) when the damper velocity is v where 0≦v≦V₀.

An example of implementation of the foregoing will now be detailed withrespect to FIGS. 3 and 4, wherein FIG. 3 is a graph 300 of a plot 302 ofsuspension spring normal force versus wheel center displacement for arepresentative motor vehicle; and FIG. 4 is a graph 400 of progressiveoptimal unconstrained damper force Φ(y) versus wheel center verticalvelocity, plot 402, for the representative motor vehicle of FIG. 3.

Given the wheel assembly mass m and the velocity U₀, the travel length Lcan be determined from the kinetic energy constraint of equation (4) asfollows: A graph of the product of spring displacement x timessuspension spring force F(x) (i.e., xF(x)) versus spring displacement xfor the predetermined spring 204 is plotted. The point on the x axis ofthe plot whereat the xF(x) axis equals

$\frac{m*U_{0}^{2}}{2}$corresponds to the predetermined travel length L where L≦L_(MAX) whereinU₀ is chosen such that L≦L_(MAX). Then F(L) can be ascertained from agraph (as per FIG. 3) of a plot of suspension spring force F(x) versusspring displacement x for the predetermined spring 204. The quantity

$\left\lbrack {\left( {1 - \frac{y^{2}}{U_{0}^{2}}} \right)L} \right\rbrack$in equation (3) can be evaluated for a velocity y, where 0≦y≦U₀, bywhich the suspension spring force

$F\left( {\left( {1 - \frac{y^{2}}{U_{0}^{2}}} \right)L} \right)$of the predetermined spring 204 can be obtained from the graph ofsuspension spring force F(x) versus spring displacement x of thepredetermined spring (i.e., FIG. 3). The progressive optimalunconstrained damper force Φ(y) as a function of wheel center velocity yof the damper 206 can now be determined from equation (3). A plot of theprogressive optimal unconstrained damper force Φ(y) versus y cansubsequently be obtained and plotted using equation (3) for variousvalues of y.

Alternatively to the immediately above paragraph, given a travel lengthL, F(L) can be ascertained from a graph (as per FIG. 3) of a plot ofsuspension spring force F(x) versus spring displacement x for thepredetermined spring 204. Velocity U₀ can be determined from equation(4). The quantity

$\left\lbrack {\left( {1 - \frac{y^{2}}{U_{0}^{2}}} \right)L} \right\rbrack$in equation (3) can be evaluated for a velocity y, where 0≦y≦U₀, bywhich the suspension spring force

$F\left( {\left( {1 - \frac{y^{2}}{U_{0}^{2}}} \right)L} \right)$of the predetermined spring 204 can be obtained from the graph ofsuspension spring force F(x) versus spring displacement x of thepredetermined spring (i.e., FIG. 3). The progressive optimalunconstrained damper force Φ(y) as a function of wheel center velocity yof the damper 206 can now be determined from equation (3). A plot of theprogressive optimal unconstrained damper force Φ(y) versus y cansubsequently be obtained and plotted using equation (3) for variousvalues of y.

For example, in FIG. 4 m=55.5 kg, L_(MAX)=0.095 m, L=0.081 m, V₀=2.7m/s, and r=0.65 from which U₀=2.7/0.65 m/s=4.1538 m/s. From point 304 ofFIG. 3, F(L) is, approximately, 5.9 kN for L=0.081 m corresponding topoint 404 of FIG. 4, where U₀=4.1538 m/s which agrees with equation (3)where Φ(U₀)=F(L). For a wheel center velocity of, for example, y=2 m/s,the quantity

$\left\lbrack {\left( {1 - \frac{y^{2}}{U_{0}^{2}}} \right)L} \right\rbrack = {0.062\mspace{14mu}{and}{\mspace{11mu}\;}{F\left( {\left( {1 - \frac{y^{2}}{U_{0}^{2}}} \right)L} \right)}}$from point 306 of FIG. 3 is, approximately, 2.8 kN. The progressiveoptimal unconstrained damper force Φ(y) from equation (3) is calculatedto be, approximately, (5.9−2.8) kN=3.1 kN whereby point 406 of FIG. 4 isobtained. Subsequent points of plot 402 can be similarly obtained forvarious values of y.

FIG. 5 is a graph 500 of damper force versus damper velocity for therepresentative motor vehicle of FIGS. 3 and 4, showing a first plot 502of damping force for a conventional passive damper, and a second plot504 of progressive optimal unconstrained damper force Ψ₁(v) as afunction of damper velocity v.

Given FIG. 4, Ψ₁(v), plot 504, can be determined from equation (5). Forexample, at point 406 of FIG. 4, Φ(y) is, approximately, 3.1 kN and y=2m/s by which v=y*r=2*0.65 m/s=1.3 m/s. From equation (5), Ψ₁(v)=3.1/0.65kN=4.8 kN when v=1.3 m/s, whereby point 506 of FIG. 5 is obtained.Subsequent points of plot 504 can be similarly obtained for variousvalues of y or v.

Ψ₁(v), plot 504, can also be determined from equation (6). For example,for L=0.081 m, F(L) is, approximately, 6.1 kN from FIG. 3. For V₀=2.7m/s and

${v = {1.3\mspace{14mu} m\text{/}s}},\left( {1 - \frac{v^{2}}{V_{0}^{2}}} \right)$L = 0.062  mand F(0.062)=2.8 kN from FIG. 3. From Equation (6), with r=0.65, Ψ₁(v)is calculated to be 4.7 kN whereby point 506 of FIG. 5 is obtained.Subsequent points of plot 504 can be similarly obtained for variousvalues of y or v.

FIG. 6 is a flow chart of an algorithm 600 for progressive optimalunconstrained damper force Φ(y) or Ψ₁(v). Algorithm 600 begins at Block602 and then proceeds to Block 604 whereat the predetermined parametersare obtained. The predetermined parameters include, but are not limitedto, m, L_(MAX), r, predetermined spring 204, and V₀ (or U₀, wherein itis understood that U₀=V₀/r) or L. Control then passes to Block 606,which uses equation (4) to determine unknown V₀ or L, whereat F(L) isdetermined from L from Block 604 using the known suspension spring forceversus displacement plot of the predetermined spring 204 as previouslydescribed. Control then passes to Block 608 whereat the progressiveoptimal unconstrained damper force Φ(y) is calculated and plotted usingequation (3) as previously described. Control then passes to Block 610whereat the progressive optimal unconstrained damper force Ψ₁(v) iscalculated and plotted using equation (5) or (6) as previouslydescribed. Control then passes to Block 612 whereat algorithm 600 ends.

As previously mentioned, in practice, a predetermined damper force φ(y)of damper 206 acting on the wheel center C_(w) below a wheel centervelocity u₁, approximately 2.0 m/s, is based on ride and handlingconsiderations for a given vehicle or vehicle model as is standard inthe art, and should not be altered. The unconstrained progressiveoptimal damper force Φ(y) obtained from equation (3), describedpreviously, requires some modifications to yield a progressive optimalconstrained event damping function Ω(y), whereby the predetermineddamper force φ(y) of the damper 206 acting on the wheel center C_(w)below a wheel center velocity of u₁, approximately 2.0 m/s, is notaltered.

If the total force, F(x)+Φ₁(y), acting on the wheel center C_(w) is aconstant equal to C₁, then the following condition applies:F(x)+Φ₁(y)≡C₁=constant   (7)by which a smooth, continuous, and monotonically increasing progressiveoptimal constrained damper force Φ₁(y) of the damper 206, as a functionof the wheel center initial position x₀ and the wheel center velocity y,can be expressed as:

$\begin{matrix}{{{\Phi_{1}(y)} = {C_{1} - {F\left( {{\frac{U_{0}^{2} - y^{2}}{2C_{1}}m} + x_{0}} \right)}}},{y \geq u_{2}}} & (8)\end{matrix}$where x(0)=x₀≦L≦L_(MAX) {dot over (x)}(0)=U₀, {dot over (x)}(t₁)=u₂, andy={dot over (x)}. F(x) in equation (7) is the suspension spring force ofthe predetermined spring 204 acting on the wheel center C_(w) for aspring displacement x, C₁ is a constant total force acting on the wheelcenter, and u₂ is an empirically determined velocity of the wheelcenter, at time t=t₁>0, greater than, but neighboring, u₁. As anexample, if u₁ is 2.0 m/s, then u₂ may be 2.69 m/s.

Velocity u₂ is empirically determined such that the transition from thepredetermined damper force φ(y) at a velocity u₁ to the progressiveoptimal constrained damper force Φ₁(y) at a velocity u₂ is a dampingforce produced by a damping force transition function. In practice, thedamping force transition function is smooth, continuous, andmonotonically increasing from u₁ to u₂ and, preferably, approximates astep function. The closer u₂ is to u₁ the better the approximation to astep function and the lower the total load on the sprung mass 202.However, u₂ should not be chosen too close to u₁ in order to avoid anabrupt change in the event damping function Ω(y) (to be describedlater), which in turn may increase loads on the sprung mass 202 forsmaller potholes than the “severe event” pothole.

Thus, the progressive optimal constrained event damping function Ω(y) asa function of wheel center velocity has the following form:

$\begin{matrix}{{\Omega(y)} = \left\{ \begin{matrix}{{{\Phi_{1}(y)} \equiv {C_{1} - {F\left( {{\frac{U_{0}^{2} - y^{2}}{2C_{1}}m} + x_{0}} \right)}}},} & {y \geq u_{2}} \\{{{step}\left( {y,u_{1},{\varphi(y)},u_{2},{\Phi_{1}(y)}} \right)},{u_{2} > y > u_{1}}} & \; \\{\mspace{95mu}{{\varphi(y)},{u_{1} \geq y \geq 0}}} & \;\end{matrix} \right.} & (9)\end{matrix}$where step is a damping force transition function having a smooth,continuous, and monotonically increasing transition from φ(y) atvelocity u₁ to Φ₁(y) at velocity u₂. Practically, the Haversine stepfunction with a cubic polynomial, well known in the art, is, preferably,used as the damping force transition function.

A progressive optimal constrained event damping function Ψ(v) as afunction of damper velocity v can be expressed as:

$\begin{matrix}{{\Psi(v)} = {\frac{\Omega\left( {y = {v/r}} \right)}{r}.}} & (10)\end{matrix}$

The constant total force C₁ (or constant acceleration C=C₁/m) isdetermined using the following procedure, per the algorithm 700 of FIG.7A, wherein the equation of motion of equation (1) is numerically solvedin conjunction with equation (9) for a determined u₂, and a minimizationof the sprung mass load is determined for a time at which {dot over(x)}=0 which corresponds to C₁:

At Block 702, equations (2) through (4) are used to determine F(L) forthe case of progressive optimal unconstrained damper force as previouslydescribed.

Next, at Block 704, F(L) is varied over an empirically determined rangeto obtain a C_(1MAX) and a C_(1MIN), for example vary F(L) by plus andminus 10% to obtain C_(1MAX)=F(L)+0.1 F(L) and C_(1MIN)=F(L)−0.1 F(L).

Next, at Block 706, a table is created of the variation of F(L) of Block704, consisting of q values wherein the first entry is designatedC₁₁=C_(1MAX), the last value is designated C_(1q)=C_(1MIN), an arbitraryentry is designated C_(1j), and adjacent values are separated by anempirically determined amount, for example 50 N.

At Block 708, each value in the table of Block 706 is set, starting withC₁₁=C_(1MAX) and ending with C_(1q)=C_(1MIN), equal to −m{dot over (x)}in equation (1) and numerically solved using equation (1) in conjunctionwith equation (9) using a particular u₂ for the time at which {dot over(x)}=0 or y=0 (i.e., y={dot over (x)}) at which time x corresponds tothe travel length of the wheel assembly and F(x) corresponds to the loadon the sprung mass 202 at full jounce for that value.

In a first alternative following Block 708, at Block 710, the solvedvalue corresponding to a minimum load on the sprung mass 202 at fulljounce is designated as C₁ and the travel length x determined for thisentry is the determined travel length L≦L_(MAX) from which F(L) may beobtained from the graph of suspension spring force F(x) versus springdisplacement x of the predetermined spring 204 (i.e., FIG. 3).

In a second alternative following Block 708, at Block 712, the load onthe sprung mass 202 at full jounce for each value in the table of Block706, starting with C₁₁=C_(1MAX) and ending with C_(1q)=C_(1MIN), isplotted versus C₁ (or C, where C=C₁/m) wherein the point on the plotwhereat a minimum load on the sprung mass 202 at full jounce occursdesignates C₁ and the travel length x determined for this entry is thedetermined travel length L≦L_(MAX) from which F(L) may be obtained fromthe graph of suspension spring force F(x) versus spring displacement xof the predetermined spring 204 (i.e., FIG. 3).

FIG. 7B depicts an example of a graph 740 of exemplar plots pursuant tothe algorithm of FIG. 7A wherein C=C₁/m and, for example m=55.5 kN. Forplot 742, if u₂=2.31 m/s, then C₁ is found at point 742 a, whereatC=108.1 m/s² and L=0.080 m. For plot 744, if u₂=2.69 m/s, then C₁ isfound at point 744 a, whereat C=110.2 m/s² and L=0.081 m. For plot 746,if u₂=3.08 m/s, then C₁ is found at point 746 a, whereat C=113.4 m/s²and L=0.081 m. Other plots for different u₂ would be similarlyevaluated.

Given x₀, r, V₀ or U₀, the wheel assembly m, and C₁, the suspensionspring force

$F\left( {{\frac{U_{0}^{2} - y^{2}}{2C_{1}}m} + x_{0}} \right)$of the predetermined spring 204 can now be determined for any y≧u₂ fromthe suspension spring force versus displacement plot of thepredetermined spring, as for example the plot of FIG. 3. The progressiveoptimal constrained damper force Φ₁(y) can then be obtained for anyy≧u₂. Thus, knowing φ(y), the step damping force transition function,and the progressive optimal constrained damper force Φ₁(y), then theprogressive optimal constrained event damping function Ω(y) as afunction of wheel center velocity y of equation (9) can be obtained forany y where 0≦y≦U₀ by which the progressive optimal constrained eventdamping function Ψ(v) as a function of damper velocity v of equation(10) can be obtained for any v where 0≦v≦V₀.

FIG. 7C is a flow chart of an algorithm 750 for a progressive optimalconstrained event damping function Ω(y). Algorithm 750 begins at Block752 and then proceeds to Block 754 whereat the predetermined parametersare obtained. The predetermined parameters include, but are not limitedto, mass m of the wheel assembly 216, L_(MAX), r, the predeterminedspring 204, U₀ or V₀, the step damping force transition function, φ(y),u₁, and x₀.

Control then passes to Block 756 whereat C₁ and u₂ are determined aspreviously described. Control then passes to Block 758 whereat theprogressive optimal constrained damper force Φ₁(y) as a function ofwheel center velocity is calculated as previously described and theprogressive optimal constrained event damping function Ω (y) as afunction of wheel center velocity is determined from equation (9).Control then passes to Block 760 whereat the progressive optimalconstrained event damping function Ψ(v) as a function of damper velocityis determined from equation (10). Control then passes to Block 762whereat algorithm 750 ends.

FIG. 8 is a graph 800 of damper force versus damper velocity for therepresentative motor vehicle of FIG. 3, showing a first plot 802 ofdamping for a conventional passive damper, and a second plot 804 ofprogressive optimal constrained damping. In FIG. 8, m=55.5 kg, r=0.65,C₁=F(L)=6116 N, C=110.2 m/sec², L=0.081 m, v₁=1.3 m/s, v₂=1.75 m/s, andV₀=2.7 m/s. The predetermined damper force φ(y) is denoted by plotportion 806 of plot 802 extending from the origin, point 808, to point810 at which the damper velocity v₁ is 1.3 m/s and wheel center velocityu₁ is 1.3/0.65=2.0 m/s. Fourth plot 814 is the step transition functionof equation (9) from point 810 to point 812 at which the damper velocityv₂ is 1.75 m/s and the wheel center velocity u₂ is 1.75/0.65=2.69 m/s.The velocity u₂ is determined as previously described. The previouslymentioned Haversine step function with a cubic polynomial is used as thetransition function from point 810 to point 812.

FIG. 9 is a graph 900 of time versus total suspension load for therepresentative motor vehicle of FIG. 3, showing a first plot 902 of asimulated suspension load having conventional passive damping; a secondplot 904 of a simulated suspension load having progressive optimalconstrained damping; and a third plot 906 of a simulated suspension loadhaving progressive optimal unconstrained damping. Point 908 depicts theexperimental peak total suspension load using a conventional prior artdamper. Point 910 depicts the experimental peak total suspension loadusing the progressive optimal constrained damping of equation (9).

FIG. 10 is a diagrammatic view 1000 of a quarter car model motor vehiclesuspension system in equilibrium (nominal trim), typically used in theart, depicting relationships, as partially presented in FIG. 2, ofpredetermined sprung mass M 202 (i.e., the vehicle body), predeterminedunsprung mass m 216 (i.e., the wheel assembly), nonlinear predeterminedfirst spring 204, nonlinear damper 206 (i.e., shock absorber, etc.), ajounce bumper 208, a nonlinear predetermined second spring 1002 (i.e.,tire) whereby F_(T)(x) is the tire spring force due to thecompressibility of the tire acting on the predetermined unsprung mass m216, and predetermined road profile 1004 (i.e., potholes). Thedisplacement x₂ of the body 202 is measured with respect to theequilibrium position 1006 and the displacement x_(g) of the roaddisturbance (i.e., pothole) 1008 is measured with respect to theequilibrium position 1010.

The predetermined road profile 1004 consists of a multiplicity ofdifferent sized potholes wherein a given sized pothole generates a givenpeak initial wheel center velocity or peak initial damper velocity for apredetermined forward velocity of the vehicle, for example 25 mph, aspreviously described. Each pothole of the multiplicity of differentsized potholes may be, for example, assigned a distinct scale factorcorresponding to its severity and peak initial wheel center velocity orpeak initial damper velocity that it generates for a predeterminedforward velocity of the vehicle, for example 25 mph. Table 1 is anexample of such pothole scaling and peak initial damper velocities,wherein each peak initial wheel center velocity equals eachcorresponding peak initial damper velocity divided by the ratio r aspreviously described, wherein, for example, r=0.68.

TABLE 1 Pothole Peak Initial Damper Pothole Scale Factor Velocity, V₀(m/s) 1 1 3.0 2 0.8 2.7 3 0.75 2.5 4 0.6 2.2 5 0.5 1.8 6 0.4 1.5

The equations of motion describing the dynamics of the quarter car modelhave the following form:m{umlaut over (x)}=−F(x−x ₂)−Ω₁({dot over (x)}−{dot over (x)} ₂)−mg−F_(T)(x−x _(g)), x(0)=x ₀ , {dot over (x)}(0)=U ₀   (11)M{umlaut over (x)} ₂ =F(x−x ₂)+Ω₁({dot over (x)}−{dot over (x)} ₂)−Mg,x(0)=x ₀ , {dot over (x)}(0)=U ₀   (12)Load=F(x−x ₂)−Ω₁({dot over (x)}−{dot over (x)} ₂).   (13)wherein the load on the sprung mass M is denoted “Load”. Ω₁ represents aprogressive optimal constrained event damping function for apredetermined pothole resulting in a predetermined peak initial wheelcenter velocity U₀. Ω₁ is determined through the solution of equations11 through 13 through the requirement that the load on the sprung massM, “Load”, is to be minimized. However, unlike the solution for Ω, asgiven by equation 9, for the equations of motion using the 1DOF motorvehicle suspension model, an analytical solution for Ω₁ is notavailable. Numerical optimization techniques, well known in the art, areutilized to determine Ω₁ for a predetermined pothole resulting in apredetermined peak initial wheel center velocity U₀. For time effectivenumerical optimization resulting in a fast convergence for the solutionfor Ω₁, the progressive optimal constrained event damping function Ωhaving the same predetermined peak initial wheel center velocity U₀ asΩ₁ is utilized as an initial event damping function Ω₁ in equations 11through 13. Examples of progressive optimal constrained event dampingfunctions obtained in this manner for the quarter car model arepresented in FIG. 15.

FIG. 11 is an exemplary diagram 1100 of potholes 1102, 1104, and 1106and corresponding sizes, wherein line 1108 depicts a level flat roadsurface. Potholes 1102, 1104, and 1106 correspond to pothole scalefactors of 1, 0.8, and 0.6, respectively, of Table 1.

FIG. 12 is a graph 1200 of a plot 1202 of tire spring normal forceversus wheel center displacement for a representative motor vehicle.Negative forces denote tire compression and negative distances denotedistances below the equilibrium position 1010.

FIG. 13A is a flow chart 1300 of an algorithm to obtain a progressiveoptimal constrained events damping function according to a preferredaspect of the present invention, utilizing the 1DOF nonlinear mechanicalsystem model.

The algorithm starts at Block 1302 and proceeds to Block 1304. At Block1304 a pothole is chosen whereby a peak initial wheel center velocity isdetermined, as, for example, depicted in Table 1. Control passes fromBlock 1304 to Block 1306 whereat a progressive optimal constrained eventdamping function is determined for the selected pothole and peak initialwheel center velocity using the 1DOF motor vehicle suspension systemmodel as previously described. Control passes from Block 1306 to Block1308, wherein if another pothole is to be chosen, then control passes toBlock 1304. Otherwise, control passes to Block 1310, whereat a lowenvelope curve is obtained, as described below with respect to FIG. 14,after which a progressing optimal events damping function is obtained,according to a preferred aspect of the present invention. Control passesfrom Block 1312 to Block 1314, whereat the algorithm ends.

FIG. 13B is a flow chart 1350 of an algorithm to obtain a progressiveoptimal constrained events damping function according to a mostpreferred aspect of the present invention, utilizing the quarter carnonlinear mechanical system model.

The algorithm starts at Block 1352 and proceeds to Block 1354. At Block1354 a pothole is chosen whereby a peak initial wheel center velocity isdetermined, as, for example, depicted in Table 1. Control passes fromBlock 1354 to Block 1356, whereat a progressive optimal constrainedevent damping function is determined for the selected pothole and peakinitial wheel center velocity using the quarter car model as previouslydescribed. Control passes from Block 1356 to Block 1358, wherein ifanother pothole is to be chosen, then control passes to Block 1354.Otherwise, control passes to Block 1360, whereat a low envelope curve isobtained, as described below with respect to FIG. 15, afterwhich aprogressing optimal events damping function is obtained, according to amost preferred aspect of the present invention. Control passes fromBlock 1362 to Block 1364, whereat the algorithm ends.

FIG. 14 is an example of a graph 1400 of damper force versus dampervelocity, depicting the construction of the low envelope curve 1402according to the preferred aspect of the present invention, utilizing a1DOF nonlinear mechanical system model.

The low envelope curve 1402 in conjunction with the predetermined damperforce φ(y) of FIG. 8 denoted by plot portion 806 of plot 802 extendingfrom the origin, point 808, to point 810 at which the damper velocity v₁is 1.3 m/s and wheel center velocity u₁ is 1.3/0.65=2.0 m/s, previouslydescribed, constitutes the progressive optimal constrained eventsdamping function according to the present invention. Plots 1404, 1406,1408, and 1410 are progressive optimal constrained event dampingfunctions generated utilizing the 1DOF nonlinear mechanical system modelhaving peak initial damper velocities of 3.5 m/s, 2.7 m/s, 2.0 m/s, and1.5 m/s, respectively, at points 1412, 1414, 1416, and 1418,respectively. The low envelope curve 1402 is generated by constructing acurve passing through the peak initial damper velocities at points 1412,1414, 1416, and 1418, respectively, and extends from point 1412 to point1420, whereat the damper velocity is 1.3 m/s. The low envelope curve1402 is incorporated with the predetermined damper force φ(y) of FIG. 8,as previously described, thereby producing the progressive optimalconstrained events damping function according to the present invention.

FIG. 15 is an example of a graph 1500 of damper force versus dampervelocity, depicting the construction of the low envelope curve 1502according to the most preferred aspect of the present invention,utilizing the quarter car nonlinear mechanical system model.

The low envelope curve 1502 in conjunction with the predetermined damperforce φ(y) of FIG. 8 denoted by plot portion 806 of plot 802 extendingfrom the origin, point 808, to point 810 at which the damper velocity v₁is 1.3 m/s and wheel center velocity u₁ is 1.3/0.65=2.0 m/s, previouslydescribed, constitutes the progressive optimal constrained eventsdamping function according to the present invention. Plots 1504, 1506,1508, 1510, and 1512 are progressive optimal constrained event dampingfunctions generated utilizing the quarter car nonlinear mechanicalsystem model having peak initial damper velocities of 2.7 m/s, 2.5 m/s,2.2 m/s, 1.8 m/s, and 1.5 m/s, respectively, at points 1514, 1516, 1518,1520, and 1522, respectively. The low envelope curve 1502 is generatedby constructing a curve passing through the peak initial dampervelocities at points 1514, 1516, 1518, 1520, and 1522, respectively, andextends from point 1514 to point 1524, whereat the damper velocity is1.3 m/s. The low envelope curve 1502 is incorporated with thepredetermined damper force φ(y) of FIG. 8, as previously described,thereby producing the progressive optimal constrained events dampingfunction according to the present invention.

FIG. 16A is a graph 1600 of force versus time, showing a first plot 1602of suspension load, a second plot 1604 of damper force, and a third plot1606 of total force of a conventional passive damping system accordingto the prior art wherein the peak total force is, approximately, 55 kNat point 1608 due to pothole 1 of Table 1.

FIG. 16B is a graph 1650 of force versus time, showing a first plot 1652of suspension load, a second plot 1654 of damper force, and a third plot1656 of total force of a progressive optimal constrained events dampingfunction according to the present invention wherein the peak total forceis, approximately, 34 kN at point 1658 due to pothole 1 of Table 1.

As used herein, by the term a “constant total force” as applied to theforce collectively provided by the spring and the damper acting on thewheel assembly during jounce according to the method of the presentinvention is meant a force in the general neighborhood of being constantincluding being exactly constant, i.e., being substantially oressentially constant.

The present invention can be implemented by adjusting the dampingresponse of any suitable damper responsive to the obtained progressiveoptimal constrained events damping function, as for preferred example byadjusting the damper disclosed in U.S. patent application Ser. No.12/238,078, filed on Sep. 25, 2008 to inventors William Golpe, ChandraS. Namuduri, Walter Cwycyshyn, and Nikolai K. Moshchuk, the disclosureof which patent application is hereby herein incorporated by reference,or by nonlimiting further example, the damper disclosed in U.S. Pat. No.5,706,919, to Kruckemeyer et al, issued on Jan. 13, 1998 to the assigneehereof, the disclosure of which patent is hereby herein incorporated byreference.

From the foregoing description, it is seen that the method according tothe present invention enables the synthesis of a non-linear compressiondamping curve to more effectively control the suspension behavior whiledriving over roads that generate maximum wheel displacements, whilemaintaining good ride quality on normal roads. Advantageously, thepresent invention provides: 1) progressive damping (by simulation andvehicle tests) to be an effective method for reducing structural loadand wheel travel at a multiplicity of high wheel velocity events (suchas potholes); and, 2) an analytical approach based on the quarter carnonlinear mechanical system model, which can be used for generating theoptimal compression damping curve that can be subsequently tuned forvehicle production.

To those skilled in the art to which this invention appertains, theabove described preferred embodiment may be subject to change ormodification. Such change or modification can be carried out withoutdeparting from the scope of the invention, which is intended to belimited only by the scope of the appended claims.

The invention claimed is:
 1. A method for providing a progressiveoptimal constrained damping response of a damper to a wheel assemblywith respect to a sprung body of a vehicle, applicable to a multiplicityof jounce events based upon a progressive optimal constrained eventsdamping function, comprising the steps of: determining a first plot of apredetermined damping force acting on the wheel center of the wheelassembly applicable to wheel center velocities below substantially apredetermined wheel center velocity between the sprung body of thevehicle and the wheel assembly, u₁, the predetermined wheel centervelocity, u₁, based on ride and handling considerations for the vehicle;generating a peak initial wheel center velocity between the sprung bodyof the vehicle and the wheel assembly respective to each jounce eventfor the vehicle traveling at a predetermined forward velocity, each peakinitial wheel center velocity obtained from a table; for each jounceevent of said multiplicity of jounce events and each peak initial wheelcenter velocity respective to each jounce event, obtaining a second plotof a progressive optimal constrained event damping function generatedfrom at least one predetermined equation of motion of the wheel center,wherein conditions applied to the at least one predetermined equationinclude no initial external forces acting on the wheel assembly, aninitial displacement x₀ of the wheel center when the initial velocity ofthe wheel assembly is U₀, and a total force acting on the wheel centerbeing substantially constant during deceleration of the wheel centerfrom the velocity U₀ to a predetermined wheel center velocity u₂,wherein the substantially constant total force is related to adetermined travel length of the wheel center such that when the wheelcenter is at the determined travel length, the wheel assembly velocityis zero, the first damper force is zero and the suspension spring iscompressed the determined travel length by which the suspension springforce is equal to the substantially constant total force and when thewheel center is at displacement x₀ the wheel assembly velocity is U₀ andthe suspension spring is compressed by x₀; constructing a low envelopecurve passing through each peak initial wheel center velocity of thesecond plot for each respective jounce event; combining the low envelopecurve with said first plot to obtain the progressive optimal constrainedevents damping function applicable to the multiplicity of jounce events;and adjusting the damping response of the damper to the wheel assemblywith respect to the sprung body of the vehicle responsive to saidprogressive optimal constrained events damping function.
 2. The methodof claim 1, wherein u₁ is substantially equal to 2.0 m/s.
 3. The methodof claim 1, wherein said equations of motion are derived from a onedegree of freedom nonlinear mechanical system model.
 4. The method ofclaim 3, wherein a first wheel assembly velocity is defined as u₁, and asecond wheel velocity is defined as u₂, wherein u₂>u₁, and wherein asecond damper force is defined as Φ₁(y), and the suspension spring forceis defined as F(x), said method further comprising the step of:determining parameters comprising: mass, m, of the wheel assembly;initial velocity, U₀, of the wheel assembly; and position x₀ of thewheel assembly relative to the predetermined reference when at U₀;wherein${{F(x)} = {F\left( {{\frac{U_{0}^{2} - y^{2}}{2C_{1}}m} + x_{0}} \right)}},$and wherein for any wheel assembly velocity, y, relative to thepredetermined reference, wherein y≧u₂, said parameters are related by:${\Phi_{1}(y)} = {C_{1} - {{F\left( {{\frac{U_{0}^{2} - y^{2}}{2C_{1}}m} + x_{0}} \right)}.}}$5. The method of claim 4, wherein a smooth and continuous damping forcetransition is provided between the first damper force at the first wheelassembly velocity u₁ and the second damper force at the second wheelassembly velocity u₂.
 6. The method of claim 5, wherein a progressiveoptimal constrained event damping function of the damper is defined asΩ(y) and has the form: ${\Omega(y)} = \left\{ \begin{matrix}{{{\Phi_{1}(y)} \equiv {C_{1} - {F\left( {{\frac{U_{0}^{2} - y^{2}}{2C_{1}}m} + x_{0}} \right)}}},} & {y \geq u_{2}} \\{{{step}\left( {y,u_{1},{\varphi(y)},u_{2},{\Phi_{1}(y)}} \right)},{u_{2} \geq y \geq u_{1}}} & \; \\{\mspace{76mu}{{\varphi(y)},{u_{1} \geq y \geq 0}}} & \;\end{matrix} \right.$ where step is a damping force transition functionhaving a smooth and continuous transition from φ(y) at velocity u₁ toΦ₁(y) at velocity u₂; and wherein for a predetermined ratio, r, suchthat said y=v/r, and v is a velocity of the damper relative to thepredetermined reference, a progressive optimal constrained event dampingfunction Ψ(v) as a function of the damper velocity v is given by therelation ${\Psi(v)} = {\frac{\Omega\left( {y = {v/r}} \right)}{r}.}$ 7.The method of claim 6, wherein x is position of the wheel assemblyrelative to the predetermined reference, and {dot over (x)}=y; andwherein the predetermined substantially constant total force C₁ isdetermined for any y≧u₂ comprising the steps of: numerically solving anequation of motion of the wheel assembly relative to the sprung mass fora predetermined range of sprung mass loads in conjunction with Ω(y) fora determined u₂, wherein the equation of motion comprises m{umlaut over(x)}+F(x)+Φ({dot over (x)})=0, x(0)=x₀, {dot over (x)}(0)=U₀; anddetermining a minimization of the sprung mass load for a time at which{dot over (x)}=0 which corresponds to C₁.
 8. The method of claim 7,wherein u₁ is substantially equal to 2.0 m/s.
 9. The method of claim 1,wherein said equations of motion are derived from a quarter carnonlinear mechanical system model.
 10. The method of claim 9, whereinthe equations of motion are of the form:m{umlaut over (x)}=−F(x−x ₂)−Ω₁({dot over (x)}−{dot over (x)} ₂)−mg−F_(T)(x−x _(g)), x(0)=x ₀ , {dot over (x)}(0)=U ₀;M{umlaut over (x)} ₂ =F(x−x ₂)+Ω₁({dot over (x)}−{dot over (x)} ₂)−Mg,x(0)=x ₀ , {dot over (x)}(0)=U ₀; andLoad=F(x−x ₂)−Ω₁({dot over (x)}−{dot over (x)} ₂); wherein M representsthe sprung mass, Load represents a load on the sprung mass, Ω₁represents a progressive optimal constrained event damping function fora predetermined pothole resulting in a predetermined peak initial wheelcenter velocity U₀; wherein Ω₁ is determined through the solution of theequations with the requirement that the Load is minimized; and whereinnumerical optimization techniques are utilized to determine Ω₁ for apredetermined pothole resulting in a predetermined peak initial wheelcenter velocity U₀.
 11. The method of claim 10, wherein u₁ issubstantially equal to 2.0 m/s.
 12. The method of claim 11, wherein saidnumerical optimization comprises: firstly utilizing a progressiveoptimal constrained event damping function Ω derived from a one degreeof freedom nonlinear mechanical system model having the samepredetermined peak initial wheel center velocity U₀ as Ω₁ in theequations of motion.
 13. A method for providing a progressive optimalconstrained damping response of a damper to a wheel assembly withrespect to a sprung body, applicable to a multiplicity of jounce eventsbased upon a progressive optimal constrained events damping function, amodel generated with respect to a wheel center of the wheel assemblyconfigured to execute the method comprising the steps of: determining afirst plot of a predetermined damping force for acting on a wheel centerof a wheel assembly applicable to wheel center velocities belowsubstantially a predetermined wheel center velocity between the sprungbody of the vehicle and the wheel assembly, u₁, the predetermined wheelcenter velocity, u₁, based on ride and handling considerations for thevehicle; generating a peak initial wheel center velocity between thesprung body of the vehicle and the wheel assembly respective to eachjounce event for the vehicle traveling at a predetermined forwardvelocity, each peak initial wheel center velocity obtained from a table;for each jounce event of said multiplicity of jounce events and eachpeak initial wheel center velocity respective to each jounce event,obtaining a second plot of a progressive optimal constrained eventdamping function generated from one degree of freedom nonlinearmechanical system model equations of motion of the wheel center, whereinconditions applied to the equations include no initial external forcesacting on the wheel assembly, an initial displacement x₀ of the wheelcenter when the initial velocity of the wheel assembly is U₀, and atotal force acting on the wheel center being substantially constantduring deceleration of the wheel center from the velocity U₀ to apredetermined wheel center velocity u₂, wherein the substantiallyconstant total force is related to a determined travel length of thewheel center such that when the wheel center is at the determined travellength, the wheel assembly velocity is zero, the first damper force iszero and the suspension spring is compressed the determined travellength by which the suspension spring force is equal to thesubstantially constant total force and when the wheel center is atdisplacement x₀ the wheel assembly velocity is U₀ and the suspensionspring is compressed by x₀; constructing a low envelope curve passingthrough each peak initial wheel center velocity of the second plot foreach respective jounce event; combining the low envelope curve with saidfirst plot to obtain the progressive optimal constrained events dampingfunction applicable to the multiplicity of jounce events; and adjustingthe damping response of the damper to the wheel assembly with respect tothe sprung body of the vehicle responsive to said progressive optimalconstrained events damping function.
 14. The method of claim l3, whereinu₁ is substantially equal to 2.0 m/s.
 15. A method for providing aprogressive optimal constrained damping response of a damper to a wheelassembly with respect to a sprung body, applicable to a multiplicity ofjounce events based upon a progressive optimal constrained eventsdamping function, a model generated with respect to a wheel center ofthe wheel assembly configured to execute the method comprising the stepsof: determining a first plot of a predetermined damping force for actingon a wheel center of a wheel assembly applicable to wheel centervelocities below substantially a predetermined wheel center velocitybetween the sprung body of the vehicle and the wheel assembly, u₁, thepredetermined wheel center velocity, u₁ based on ride and handlingconsiderations for the vehicle; generating a peak initial wheel centervelocity between the sprung body of the vehicle and the wheel assemblyrespective to each jounce event for the vehicle traveling at apredetermined forward velocity, each peak initial wheel center velocityobtained from a table; for each jounce event of said multiplicity ofjounce events, obtaining a second plot of a progressive optimalconstrained event damping function generated from quarter car nonlinearmechanical system model equations of motion of the wheel center, whereinconditions applied to the equations include no initial external forcesacting on the wheel assembly, an initial displacement x₀ of the wheelcenter when the initial velocity of the wheel assembly is U₀, and atotal force acting on the wheel center being substantially constantduring deceleration of the wheel center from the velocity U₀ to apredetermined wheel center velocity u₂, wherein the substantiallyconstant total force is related to a determined travel length of thewheel center such that when the wheel center is at the determined travellength, the wheel assembly velocity is zero, the first damper force iszero and the suspension spring is compressed the determined travellength by which the suspension spring force is equal to thesubstantially constant total force and when the wheel center is atdisplacement x₀ the wheel assembly velocity is U₀ and the suspensionspring is compressed by x₀; constructing a low envelope curve passingthrough each peak initial wheel center velocity of the second plot foreach respective jounce event; combining the low envelope curve with saidfirst plot to obtain the progressive optimal constrained events dampingfunction applicable to the multiplicity of jounce events; and adjustingthe damping response of the damper to the wheel assembly with respect tothe sprung body of the vehicle responsive to said progressive optimalconstrained events damping function.
 16. The method of claim 15, whereinu₁ is substantially equal to 2.0 m/s.